Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 393470,4pages doi:10.1155/2010/393470
Research Article
A Note on Geodesically Bounded R -Trees
W. A. Kirk
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
Correspondence should be addressed to W. A. Kirk,[email protected] Received 4 March 2010; Accepted 10 May 2010
Academic Editor: Mohamed Amine Khamsi
Copyrightq2010 W. A. Kirk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
It is proved that a complete geodesically boundedR-tree is the closed convex hull of the set of its extreme points. It is also noted that ifXis a closed convex geodesically bounded subset of a completeR-treeY,and if a nonexpansive mappingT:X → Ysatisfies inf{dx, Tx:x∈X}0, thenThas a fixed point. The latter result fails ifTis only continuous.
1. Introduction
Recall that for a metric spaceX, d,a geodesic pathor metric segmentjoiningxandyinX is a mappingcof a closed interval0, lintoXsuch thatc0 x, cl y,anddct, ct
|t−t|for eacht, t∈0, l.Thuscis an isometry anddx, y l.AnR-treeor metric treeis a metric spaceXsuch that:
ithere is a unique geodesic pathdenoted byx, yjoining each pair of pointsx, y∈ X;
iiify, x∩x, z {x},theny, x∪x, z y, z.
Fromiandii, it is easy to deduce that
iiiifx, y, z∈X,thenx, y∩x, z x, wfor somew∈X.
The concept of anR-tree goes back to a 1977 article of Tits1. CompleteR-trees posses fascinating geometric and topological properties. Standard examples ofR-trees include the
“radial” and “river” metrics onR2.For the radial metric, consider all rays emanating from the origin inR2.Define the radial distancedr betweenx, y ∈ R2 to be the usual distance if they are on the same ray; otherwise take
dr
x, y
dx,0 d 0, y
. 1.1
2 Fixed Point Theory and Applications Hereddenotes the usual Euclidean distance and 0 denotes the origin.For the river metricρ onR2, if two pointsx, andyare on the same vertical line, defineρx, y dx, y.Otherwise defineρx, y |x2||y2||x1−y1|,wherex x1, x2andy y1, y2.More subtle examples ofR-trees also exist, for example, the real tree of Dress and Terhalle2.
It is shown in3thatR-trees complete are hyperconvex metric spacesa fact that also follows from Theorem B of4and the characterization of5. They are also CAT0spaces in the sense of Gromovsee, e.g.,6, page 167. Moreover, complete and geodesically bounded R-trees have the fixed point property for continuous maps. This fact is a consequences of a result of Young7 see also8, and it suggests that complete geodesically boundedR- trees have properties that one often associates with compactness. The two observations below serve to affirm this.
2. A Krein-Milman Theorem
In9 Niculescu proved that a nonempty compact convex subsetX of a complete CAT0 spacecalled a global NPC space in9is the convex hull of the set of all its extreme points.
Subsequently, in10, Borkowski et al. provedamong other thingsthat compactness is not needed in the special case whenXis a complete and boundedR-tree. Here we show that in completeR-trees even the boundedness assumption may be relaxed.
Theorem 2.1. LetXbe a complete and geodesically boundedR-tree. ThenXis the convex hull of its setEof extreme points.
Proof. Letx ∈E,and letz∈X\E.We will show thatzlies on a segment joiningxto some other element ofE.We proceed by transfinite induction. LetΩdenote the set of all countable ordinals, letz0 z,letα∈Ω,and assume that for allβ∈Ωwithβ < α, zβhas been defined so that the following condition holds:
iμ < γ < α⇒zμ∈x, zγ,andzγ/∈E⇒zμ/zγ. There are two cases.
1α β1.Ifzβ ∈E,there is nothing to prove becausez z0 ∈x, zβ. Otherwise, there are elementsa, b∈Xsuch thatzβlies on the segmenta, banda /zβ/b.At least one of these points,saya,does not lie on the segmentzβ, x. Setzα a,and observe thatzβlies on the segmentzα, x.
2αis a limit ordinal. Since X is geodesically bounded, it must be the case that
β<αdzβ, zβ1<∞.This implies thatzββ<αis a Cauchy net. SinceXis complete, it must converge to somezα∈X.
Therefore, zα is defined for all α ∈ Ω. Since X is geodesically bounded,
β∈Ωdzβ, zβ1 < ∞.But sinceΩis uncountable, it is not possible that dzβ, zβ1 > 0 for eachβ.Hence this transfinite process must terminate, andzβ zβ1 for someβ ∈Ω.It now follows fromithatzβ∈Eandzlies on the segmentzβ, x.
Remark 2.2. The above proof shows that in fact each point ofX is on a segment joining any given extreme point to some other extreme point.
Fixed Point Theory and Applications 3
3. A Fixed Point Theorem
It is known that ifKis a bounded closed convex subset of a complete CAT0spaceY,and if f:K → Y is a nonexpansive mapping for which
inf d
x, fx
:x∈K
0, 3.1 then f has a fixed point see11, Theorem 21; also12, Corollary 3.8. This fact carries over toR-trees sinceR-trees are also CAT0spaces. However, we note here that ifY is an R-tree, then again boundedness ofK can be replaced by the assumption that K is merely geodesically bounded. In fact, we prove the following.In the following theorem, we assume Tis nonexpansive relative to the Hausdorffmetric on the bounded nonempty closed subsets ofY.
Theorem 3.1. SupposeXis a closed convex and geodesically bounded subset of a completeR-treeY, and supposeT :X → 2Yis a nonexpansive mapping taking values in the family of nonempty bounded closed convex subsets ofY.Suppose also that inf{distx, Tx:x∈ X} 0.Then there is a point x∈Xfor whichx∈Tx.
We will need the following result in the proof ofTheorem 3.1.See13,14for more general set-valued versions of this theorem.
Theorem 3.2. SupposeXis a closed convex geodesically bounded subset of a completeR-treeY and supposef :X → Yis continuous. Then eitherfhas a fixed point or there exists a pointz∈Xsuch that
0< d
z, fz inf
d
x, fz
:x∈X
. 3.2
Proof ofTheorem 3.1. Since complete R-trees are hyperconvex, by Corollary 1 of 15 the selectionf : X → Y defined by takingfx to be the point ofTxwhich is nearest to x for eachx ∈ X is a nonexpansive single-valued mapping. Now assume f does not have a fixed point. Then byTheorem 3.2there existsz∈Xsuch that
0< d
z, fz inf
d
x, fz
:x∈X
. 3.3
We assert thatdx, fx ≥ dz, fzfor eachx ∈X.Indeed letx ∈X.Byiiithere exists w ∈ Y such thatz, fz∩z, x z, w.But sinceX is convexz, x ⊆ X,sow ∈ z, x impliesw∈X.Alsow∈z, fz,so it follows from3.3thatwz.Thusz, fz∩z, x {z},and the segmentx, fzmust pass throughz. Therefore,
dx, z d
z, fz d
x, fz
≤d
x, fx d
fx, fz
≤d
x, fx
dx, z.
3.4
Thus inf{dx, fx:x∈X} ≥dz, fz>0 – a contradiction. Therefore, there existsx∈X such thatxfx∈Tx.
4 Fixed Point Theory and Applications Corollary 3.3. SupposeXis a closed convex and geodesically bounded subset of a completeR-treeY, and supposef:X → Yis a nonexpansive mapping for which inf{dx, fx:x∈X}0.Thenf has a fixed point.
Example 3.4. In view of the fact that continuous self-maps ofX → X have fixed points, it is natural to ask whetherCorollary 3.3 holds for continuous mappings. The answer is no, even whenX is bounded. LetY be the Euclidean plane R2 with the radial metric. Let{en} be a sequence of distinct points on the unit circle, and letX ∪∞n1en,0.We now define a continuous fixed-point free mapf:X → Yfor which inf{dx, fx:x∈X}0. First move each point of the segment0, e1to the right onto a segmente1, bwhereb /e1ande1, b is on the ray which extends0, e1.Thusf0, e1 e1, b.For eachn≥ 2,letan denote the point on the segmenten,0which has distance 1/nfromen. It is now clearly possible to construct a continuouseven lipschitzianfixed point-free mapf a shiftof the segment en,0onto the segmentan, e1, n≥ 2,for whichfen an.Thusden, fen 1/nfor all n.
Remark 3.5. Corollary 3.3for boundedXis also a consequence of Theorem 6 of15.
References
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